Probability: the basics

In short: Probability describes how certain an event is to occur, as a number between 0 (impossible) and 1 (certain). A few rules let you compute games of chance, tests and risks.

In more detail: For equally likely outcomes: probability = favourable cases divided by possible cases. A die shows a 6 with 1/6 ≈ 16.7 %.

In Depth

• Complement: P(not A) = 1 − P(A). “At least one 6 in four throws” is easier as 1 − (5/6)⁴ ≈ 51.8 %.
• And (independent): P(A and B) = P(A) · P(B). Two sixes in a row: 1/36.
• Or (mutually exclusive): P(A or B) = P(A) + P(B). If not exclusive, subtract the overlap.
• Conditional probability: P(A | B) is the probability of A given that B has already occurred. See Bayes’ theorem.
• Tree diagram: Helps with multi-stage random experiments. Path probabilities are multiplied, paths are added.
• Expected value: Mean over many repetitions: sum of value times probability. A lottery ticket winning €1 at 1 % has expected value €0.01.
• Law of large numbers: With many repetitions the relative frequency approaches the probability. There is no “law of compensation”: a die does not “remember” previous throws (gambler’s fallacy).
• Applications: Insurance, games, quality control, random algorithms (Monte Carlo), language models.

See also: normal distribution